What is the half-life of an isotope if it decays to 12.5% of its radioactivity in 18 minutes?
k = 0.693/t1/2
ln(100/12.5) = k*18
Solve for k, then
k = 0.693/t(sub>1/2.
Solve for t1/2.
To find the half-life of an isotope, we need to determine the time it takes for the radioactivity to decrease to half of its initial value.
In this case, the isotope decays to 12.5% of its initial radioactivity. We can calculate the fraction of its initial radioactivity as follows:
Fraction of radioactivity remaining = 12.5% = 0.125
The fraction remaining after time t is given by:
Fraction remaining = (1/2)^(t/h)
Where t is the elapsed time and h is the half-life.
Since the fraction remaining is 0.125, we have:
0.125 = (1/2)^(t/h)
Taking the logarithm of both sides (base 2), we get:
log2(0.125) = log2((1/2)^(t/h))
Simplifying further:
-3 = t/h
Since we know that the elapsed time is 18 minutes (t = 18), we can solve for h:
-3 = 18/h
Solving for h, we get:
h = 18 / -3 = -6
However, half-life cannot be negative. Therefore, we made an error somewhere in the calculations or interpretation of the problem. Please double-check the problem statement and try again.
To find the half-life of an isotope, we need to determine the time it takes for the radioactivity to decrease by half. In this case, we are given that the isotope decays to 12.5% of its original radioactivity.
To find the half-life, we can set up the following equation:
Final radioactivity = Initial radioactivity * (1/2)^(time/half-life)
Given that the final radioactivity is 12.5% (which is equivalent to 0.125) of the initial radioactivity, we can substitute these values into the equation:
0.125 = 1 * (1/2)^(18/half-life)
Now, let's solve for the half-life.
First, isolate the exponential term:
(1/2)^(18/half-life) = 0.125
Take the natural logarithm (ln) of both sides to get rid of the exponent:
ln[(1/2)^(18/half-life)] = ln(0.125)
Apply the property of logarithms that states the natural logarithm of an exponent is equal to the exponent multiplied by the natural logarithm of the base:
(18/half-life) * ln(1/2) = ln(0.125)
Next, let's solve for (18/half-life) by dividing both sides by ln(1/2):
18/half-life = ln(0.125) / ln(1/2)
To find the half-life, multiply both sides by (half-life/18) and solve for (half-life):
half-life = 18 * ln(1/2) / ln(0.125)
Using a scientific calculator or any computer program with logarithmic capabilities, we can approximate the value of ln(1/2) and ln(0.125) to calculate the half-life of the isotope.